Differential Operators on an Affine Curve

Abstract

Let X denote an irreducible affine algebraic curve over an algebraically closed field k of characteristic zero. Denote by Dx the sheaf of differential operators on X, and D(X)=Γ(X,Dx), the ring of global differential operators on X. The following is established: THEOREM. D(X) is a finitely generated k‐algebra, and a noetherian ring. Furthermore, D(X) has a unique minimal non‐zero ideal J, and D(X)/J is a finite‐dimensional k‐algebra. Let X ˜ denoted the normalisation of X, and π: X ˜ →X the projection map. The main technique is to compare D(X) with D( X ˜ ). THEOREM. The following are equivalent: (i) π is injective, (ii) D(X) is a simple ring, (iii) D(X) is Morita equivalent to D( X ˜ ), (iv) the categories DX‐Mod and DX˜‐Mod are equivalent, (v) gr D(X) is noetherian, (vi) the global homological dimension of D(X) is 1. For higher‐dimensional varieties the techniques produce examples of varieties X for which D(X) is right but not left noetherian.

Differential operators on Stanley-Reisner rings

“…The problem now is to understand the properties of D(S) when X is singular. There are several interesting results, for example, in [2], [9] and [14,Section 7]. Now let R = k[x 1 , .…”

One-dimensional infinitesimal-birational duality through differential operators

“…It is easy to see that its composition with the classical Fourier transformation (30) gives the same as (29), which proves the commutativity of the above square.…”

“…The left hand side of this diagram is already described in Corollary 5 and Theorem 3. The composition on the left hand side is (29) k…”

Rings of differential operators on one-dimensional algebras

“…The primary purpose of this paper is to study the ring 3f[A) of differential operators on A when dim(^4), the Krull dimension of A, is at most one. If A is also reduced or is a domain 3f(A) has been studied extensively in [10] and [15] and we prove analogues of the main results of these papers. For example THEOREM A.…”